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The approximation of flows
Authors:Derek W Robinson
Institution:1. Département de Physique, Université d''Aix-Marseille Marseille, France;2. Centre de Physique Théorique, CNRS, Marseille, France
Abstract:Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space X with corresponding infinitesimal generators S, T. We prove the following: ∥Ut, ? Vt ∥ = O(t), t → 0, if and only if U = V; ∥Ut ? Vt∥ = O(tα), t → 0; with 0 ? α ? 1, if and only if S = Ω(T + P)Ω?1, where Ω, P, are bounded operators on X such that ∥UtΩ ? ΩUt∥ = O(tα), ∥UtP ? PUt∥ = ?O(tα), t → 0; ∥Ut ? Vt∥ = O(t) if and only if S1 ? T1 has a bounded extension to X1. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras.
Keywords:
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